Let's say we have a set A = {1, 2, 3, 4}
I understand that, for example, $\displaystyle \left\{{2, 3}\right\}\in P(A)$, but why is it that$\displaystyle \left\{{\varnothing}\right\}\subseteq P(A)$?
Take a subset S of A.
Then this subset is an element of P(A).
S is in P(A)
For a set to be a subset of P(A), each of its elements must be in P(A).
Ie each of its elements must be a subset of A.
The empty set is a subset of A.
The set containing the empty set is therefore a subset of P(A)